## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 12, Number 3 (2002), 1096-1113.

### On invariant measures of discrete time filters in the correlated signal-noise case

#### Abstract

The classical results on the ergodic properties of the nonlinear filter previously have been proved under the crucial assumption that the signal process and the observation noise are independent. This assumption is quite restrictive and many important problems in engineering and stochastic control correspond to filtering models with correlated signal and noise. Unlike the case of independent signal and noise, the filter process in the general correlated case may not be Markov even if the signal is a Markov process. In this work a broad class of discrete time filtering problems with signal-noise correlation is studied. It is shown that the pair process $(Y_j, \pi_j)_{j \in \mathbb{N}_0}$ is a Feller-Markov process, where $(Y_j)_{j \in \mathbb{N}_0}$ is the observation process and $\pi_j$ is the filter, that is, the conditional distribution of the signal: $X_j$ given past and current observations. It is shown that if the signal process $(X_j)$ has an invariant measure, then so does $(Y_j, \pi_j)$. Finally, it is proved that if $(X_j)$ has a unique invariant measure and the stationary flow corresponding to the signal process is purely nondeterministic, then the pair $(Y_j, \pi_j)$ has a unique invariant measure.

#### Article information

**Source**

Ann. Appl. Probab., Volume 12, Number 3 (2002), 1096-1113.

**Dates**

First available in Project Euclid: 12 September 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1031863182

**Digital Object Identifier**

doi:10.1214/aoap/1031863182

**Mathematical Reviews number (MathSciNet)**

MR1925453

**Zentralblatt MATH identifier**

1012.60044

**Subjects**

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60J05: Discrete-time Markov processes on general state spaces 60H15: Stochastic partial differential equations [See also 35R60]

**Keywords**

Nonlinear filtering invariant measures asymptotic stability measure valued processes

#### Citation

Budhiraja, A. On invariant measures of discrete time filters in the correlated signal-noise case. Ann. Appl. Probab. 12 (2002), no. 3, 1096--1113. doi:10.1214/aoap/1031863182. https://projecteuclid.org/euclid.aoap/1031863182